Given a function of two variables, say f(x,y), what are some known techniques to prove that it has multiple maxima? I can see via simulation that this is the case, but trying to figure out a formal way to do so.
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This strongly depends on what kind of a function $f$ is... Is $f$ an aribtrary function $\Bbb{R} \times \Bbb{R} \to \Bbb{R}$? Or is $f$ continuous, differentiable or even two times differentiable? – aexl May 19 '15 at 03:49
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Yes, it is a double differentiable function. R^+ \times R^+->R^+ – Dhiraj May 19 '15 at 03:53
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Given a point $p$ inside a closed loop $C$ and constant $c$ such that $f(x) \le c < f(p) $ for all $x \in C$, $f$ must have a local maximum inside $C$. This can allow you to show that $f$ has multiple local maxima without actually finding them.
Robert Israel
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